To check whether an array list contains only arrangements of colors corresponding to allowed templates one can then use SatisfiedQ[list_, allowed_] := Apply[And, Map[MatchQ[#, allowed] &, Partition[list, {3, 3}, {1, 1}], {2}], {0, 1}]

After n steps the total length of all stems is given by Apply[Plus, Abs[b]] n .

Then at each step one applies the rule {r, s} -> If[r >= s+1, {4(r–s–1), 2(s+2)}, {4r, 2s}] .

In each case the whole pattern can be generated by repeatedly applying the substitution system rule shown.

For k = 2 , the number of rules that conserve the total number of black cells can be computed from q = Binomial[n, Range[0, n]] as Apply[Times, q q ] . The number of these rules that are also reversible is Apply[Times, q!]

Proof structures The proof shown is in a sense based on very low-level steps, each consisting of applying a single axiom from the original axiom system. … And in the case shown here if one first proves the lemma (a ⊼ (a ⊼ (b ⊼ ((a ⊼ a) ⊼ c))))  (b ⊼ a) and treats it as rule 6, then the main proof can be shortened: When one just applies axioms from the original axiom system one is in effect following a single line of steps. … In the way I have set things up one always gets from one step in a proof to the next by taking an expression and applying some transformation rule to it.

A typical example of the first approach is the Ising model for spin systems in which relative probabilities of sequences are found by multiplying together the results of applying a simple function to blocks of nearby elements in the sequence.

Proofs in Mathematica Most of the individual built-in functions of Mathematica I designed to be as predictable as possible—applying transformations in definite ways and using algorithms that are never of fundamentally unknown difficulty.

Implementation [of patterning model] Given a 2D array of values a and a list of weights w , each step in the evolution of the system corresponds to WeightedStep[w_List, a_] := Map[If[# > 0, 1, 0]&, Sum[w 〚 1 + i 〛 Apply[Plus, Map[RotateLeft[a, #]&, Layer[i]]], {i, 0, Length[w] - 1}], {2}] Layer[n_] := Layer[n] = Select[Flatten[Table[{i, j}, {i, -n, n}, {j, -n, n}],1], MemberQ[#, n| - n]&]

Financial Systems During the development of the ideas in this book I have been asked many times whether they might apply to financial systems.